VOL. LXVII.] PHILOSOPHICAL TRANSACTIONS. 14g 



q'q"q"' &c. will be = \/{r^ — h'), the distance of these parallel circles being go'^. 

 Therefore their peripheries being as the velocities (v and u) with which they are 

 described, their radii h and -/ (r'^ — /r) will be in the ratio of the said velocities; 

 that is V : u :: h : n/ {r^ — h^) ; whence, " being = — ^^^' — , h, the radius of the 



circle p'p"p"' &c. is found = ., ,'\ — ^ = ^ — —•, and V {r"^ — k') the ra- 



V(« + V-) e'u^ ^ ' 



dius of the circle q'q'q" &c. = ., .^ , o-. = ri 5 ^ being = — , the v&i 



locity with which the momentary pole /)", p" , he. changes its place. Conse- 

 quently, if pr' be an arc in the said immoveable concave sphere whose sine is 

 — — r = , , , the great circles o'?', o' p", q"'v", &c. will intersect 



each other at the point r. 



7. Further, the force p being invariable, and acting as expressed in the pre- 

 ceding article, the primitive pole p' and the momentary poles p",}"', &c. will all 

 be found in a circle p'p"]i" &c. described on the surface of the revolving sphere, 

 as observed in that article; which circle, during the action of the force of f, will 

 (as is also observed in the said article) always touch and roll along the 

 immoveable circle (p'p"p'" &c.) whose radius we have just now found =1 



— ; — J z= —^ ; the point of contact being always the momentary pole. 



Let the sine of the arc p'q of the great circle rp'q^'t in the revolving sphere 

 be equal to^, the radius of the said circle p'p"p"' &c. ; then will the point q and 

 its opposite point (o) in the surface of the said sphere, during the action of the 

 force F, describe circles in the surrounding immoveable concave sphere parallel 

 to (p'p "p"' &c.) the circle described by the momentary pole p", //", &c. in the 

 same concave sphere. And such point a and its opposite point (o) being conti- 

 nually urged by the force f in directions at right angles to the tangents to the 

 arcs they describe, their velocity will continue the same as before the action of 

 the said force commenced; which velocity, and the radius of the said circle 

 p'p"p"' &c. will be determined by the following computation. 



That radius being denoted by k, we have r : k :: e : — , the velocity of the point 



a before the action of the force p commenced, and h :v ::k : -r, the velocity of 

 the same point (a) during the action of that force, k being put for the sine of 

 the arc aR; therefore, the velocity of a continuing the same during the action 

 of p as before, we have — = -r. But k is the sine of the sum of the arcs rp', 



p'a, whose sines are h and k respectively; therefore — —_ 1 — i-il-^I — ; will 



